Unit 7B

Integrals Part 2

Lesson 3: Integration

Velocity from Acceleration

In Lesson 2, the acceleration function «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»a«/mi»«mfenced»«mi»t«/mi»«/mfenced»«/math» of an object travelling along a straight line was calculated by differentiating the velocity function «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»v«/mi»«mfenced»«mi»t«/mi»«/mfenced»«/math».


Since integration is the inverse operation of differentiation, the process can be reversed to find a velocity function given an acceleration function.

Therefore, if «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»a«/mi»«mfenced»«mi»t«/mi»«/mfenced»«mo»=«/mo»«mfrac»«mrow»«mi»d«/mi»«mi»v«/mi»«/mrow»«mrow»«mi»d«/mi»«mi»t«/mi»«/mrow»«/mfrac»«/math», then «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»v«/mi»«mfenced»«mi»t«/mi»«/mfenced»«mo»=«/mo»«mo»§#8747;«/mo»«mi»a«/mi»«mfenced»«mi»t«/mi»«/mfenced»«mi mathvariant=¨normal¨» «/mi»«mi»d«/mi»«mi»t«/mi»«/math».

Recall acceleration is the second derivative of the position function «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»s«/mi»«mfenced»«mi»t«/mi»«/mfenced»«/math».


Therefore, given the acceleration function, the position function can be determined by integrating twice. The first integration yields the velocity function, and the second integration yields the position function.

Watch the video Position, Velocity, and Acceleration with Integrals to see the connection between position, velocity, and acceleration. Note the position function is denoted by «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»d«/mi»«mfenced»«mi»t«/mi»«/mfenced»«/math» in the video instead of «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»s«/mi»«mfenced»«mi»t«/mi»«/mfenced»«/math».